MuonGS: Pretraining LLMs with Relaxed Weight-Norm Control

The paper introduces MuonGS, a variant of the Muon optimizer that employs Gram–Schmidt projection to remove the radial component of updates, thereby controlling weight norm growth without fixing it. By ensuring the weight norm grows at most proportionally to √{t}, MuonGS achieves a cumulative effective learning rate of ≥√{T} and an O(T^{-1/2}) stationarity guarantee for smooth scale‑invariant objectives. Experiments on 160M‑ and 400M‑parameter dense models and a 135M‑active/1B‑total MoE model show MuonGS attains lower validation perplexity than MuonH and an estimated token‑equivalent speedup of up to ~15% at matched validation loss.

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